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    Orc in the Playground
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    Default Good book for self-study of undergraduate-level (math) analysis?

    Inspired by (and not wanting to hijack) the Boolean Algebra thread (which pointed out a neat book for Abstract Algebra that I might read through as a refresher over winter break since it's been a while since I needed it), I was wondering if anyone had some good suggestions for undergraduate-level books on Analysis?

    Background on me: I somehow never got around to taking an Analysis class in college, and it's kind of a glaring gap in my math background when I look at "things I know" versus "things I'd need to know if I wanted to take some graduate-level math classes". I've taught middle and high school math for about a decade now. I originally studied both computer science and communication as an undergrad with the idea I'd pursue HCI in grad school and become a computer science professor at a teaching-focused SLAC someday, but after I started grad school I quickly realized that I liked to talk to people much more than I liked to do research, so I switched career aspirations to middle/high school teacher to avoid spending years of time pursuing research that my heart just wasn't in.

    Now, I'd like to teach some dual-credit math classes, and in order to do that I need a certain number of graduate credits in mathematics (rather than computer science) so I can offer courses for college credit through one of our local colleges in addition to high school credit. I'd like to try to work through some undergraduate-level work on my own to get a better sense of what kinds of things one does in the undergraduate analysis sequence and what I should brush up on before taking it. (The point is to be ready to eventually take grad-level classes, so I need to both actually understand the undergrad material well and to get good grades in it. I'd prefer to front-load as much of the "understanding" part as possible to self-study before I take the class so I can better assure the "good grades" part during the actual school term, particularly if it turns out I need to review a bunch of things from other math classes that I haven't needed in a while first.)

    I generally did well in proofs/logic based courses and prefer a fairly abstract approach over a calculation-intensive one, if possible. (My worst grade in a college math class was in 200-level differential equations, because it was all cookbook stuff with a textbook-provided program we used to generate vector fields and such, and very little theory for me to hang my hat on to keep track of why I was doing anything or if my approach and/or solution to a problem made sense. My best classes were things like discrete math and abstract algebra where we proved everything in the course.)

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    Default Re: Good book for self-study of undergraduate-level (math) analysis?

    You could ask some of the math teachers at your local university. Or a non-local university, if you're worried asking might somehow come back to bite you with the dual-credit teaching thing. I imagine many professors would be happy to help someone out by giving recommendations, if they know someone is interested in graduate work but wants some refreshing/covering-their-bases first.

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    Default Re: Good book for self-study of undergraduate-level (math) analysis?

    I'd recommend Ray Mayer's course notes on analysis. It's taught for at what looks like a 1st year level. I've generally been good at math, but reading through the first few sections definitely gave me a better appreciation for it, and much more confidence in tackling new subjects. Since he specifically allows reproduction of the notes, I even took them to my university's printing services to have hard copies printed and bound.
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    Default Re: Good book for self-study of undergraduate-level (math) analysis?

    Analysis inescapably will bury you in Epsilons and Deltas just because Analysis is predicated almost entirely on the Real Line and friends as Metric Spaces.

    If you like abstraction you can ignore analysis and go into Topology. Morris "Topology without Tears" which has a free download provided by the author is a gentle introduction to the topic. Munkres is THE textbook for topology, though it is recommended one do the exercises on each section (they are not rote applications but problems you HAVE TO FACE FOR YOURSELF to build a better intuition on the topic) otherwise it can feel a bit suffocating. Pi-base and Counterexamples in Topology help a lot for finding cool places where the obvious is false.

    Why topology if you wanted analysis? Because all analysis courses are just doing topology in R and R^n. You will be doing manual proofs for results which are true for FAR MORE GENERAL objects. Topology let's you see why those results are not happy coincidences of those spaces but things that must be true because there is no other way. On the other hand topology without analysis might leave you wanting for examples in numerous cases, though if you are ok with abstraction that's irrelevant.

    If you still persist on Analysis despite knowing how much prettier, better, faster, more intuitive (not at all biased on my part) Topology is, Folland's Real Analysis is a very good introductory book on the topic, as is Understanding Analysis by Abbot. An intermediate book is Elementary Analysis by Ross.

    The one I WANT to recommend would be Rudin. But Rudin is BRUTAL. You need to respect the book and take your time. It's arguably THE BEST analysis book in terms of content but the first sections are extremely dense and will grind through your soul. If you manage to survive the first few chapters it's smooth sailing since it will imprint analysis on your soul. But that's an IF.

    Tao's Analysis is a good, less sadistic but almost as equally comprehensive as the Rudin book. I haven't gone through all of it since I had already weathered the Rudin but it's for the most part a more tractable version of the Rudin.

    Some people recommend Apostol, but I abhor his books from bad experiences on his book on Linear Algebra.
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    Default Re: Good book for self-study of undergraduate-level (math) analysis?

    Quote Originally Posted by AsteriskAmp View Post
    Analysis inescapably will bury you in Epsilons and Deltas just because Analysis is predicated almost entirely on the Real Line and friends as Metric Spaces.

    If you like abstraction you can ignore analysis and go into Topology. Morris "Topology without Tears" which has a free download provided by the author is a gentle introduction to the topic. Munkres is THE textbook for topology, though it is recommended one do the exercises on each section (they are not rote applications but problems you HAVE TO FACE FOR YOURSELF to build a better intuition on the topic) otherwise it can feel a bit suffocating. Pi-base and Counterexamples in Topology help a lot for finding cool places where the obvious is false.

    Why topology if you wanted analysis? Because all analysis courses are just doing topology in R and R^n. You will be doing manual proofs for results which are true for FAR MORE GENERAL objects. Topology let's you see why those results are not happy coincidences of those spaces but things that must be true because there is no other way. On the other hand topology without analysis might leave you wanting for examples in numerous cases, though if you are ok with abstraction that's irrelevant.

    If you still persist on Analysis despite knowing how much prettier, better, faster, more intuitive (not at all biased on my part) Topology is, Folland's Real Analysis is a very good introductory book on the topic, as is Understanding Analysis by Abbot. An intermediate book is Elementary Analysis by Ross.

    The one I WANT to recommend would be Rudin. But Rudin is BRUTAL. You need to respect the book and take your time. It's arguably THE BEST analysis book in terms of content but the first sections are extremely dense and will grind through your soul. If you manage to survive the first few chapters it's smooth sailing since it will imprint analysis on your soul. But that's an IF.

    Tao's Analysis is a good, less sadistic but almost as equally comprehensive as the Rudin book. I haven't gone through all of it since I had already weathered the Rudin but it's for the most part a more tractable version of the Rudin.

    Some people recommend Apostol, but I abhor his books from bad experiences on his book on Linear Algebra.
    Topology is great, but this doesn't really address Alegh's question. Their issue is that they're missing what is generally considered a crucial piece of higher math education, and learning a generalization of certain parts of it won't really help with the requirement-satisfying part.

    Also, analysis is not actually just the topology of R^n. It's true that you need a certain amount of topology to do analysis, so introductory courses spend a fair amount of time developing that, but most of the meat of analysis is in genuinely analytic concepts like differentiation, integration, series, and debatably uniform continuity, uniform convergence, and completeness, which do not generalize to arbitrary topological spaces.

    Furthermore, the definition of a topological space is pretty hard to motivate unless you spend a certain amount of time playing with metric spaces (or do something like this). Yes, Alegh claims to like abstraction, but there's a difference between normal person liking abstraction and the kind which lets you jump into most topology textbooks with no background.

    As far as analysis books, I used Lebl's Basic Analysis for my first course. I'm not sure it's the best book, but it's better than many and free. If you want more abstraction, read chapter 7 early.

    And once you're done with whatever book you end up reading, it might well be a good idea to take a look at topology. It's lots of fun and also an essential part of a higher math education.
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    Default Re: Good book for self-study of undergraduate-level (math) analysis?

    There are quite a few good introductory real analysis texts. This thread is doing a good job of covering those already.

    For a companion book to whatever you choose, I recommend How to Think About Analysis by Lara Alcock.

    Also, if you happen to decide to study topology as well, I highly recommend Introduction to Topology by Gamelin And Greene. Personally, I find it to be structured better than Munkres, with better explanations and motivation of concepts. I like the exercises a bit better, as well. That's all subjective on my part, though. I happen to like the form factor of the actual book, as well. It's relatively slim and easy to hold for reading. The price is low, too.
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    Default Re: Good book for self-study of undergraduate-level (math) analysis?

    Quote Originally Posted by PirateMonk View Post
    Topology is great, but this doesn't really address Alegh's question. Their issue is that they're missing what is generally considered a crucial piece of higher math education, and learning a generalization of certain parts of it won't really help with the requirement-satisfying part.
    I did list analysis books later. I will agree you can't ENTIRELY dispense with Analysis but approaching Analysis from Topology IMO is a more elegant way about it. Analysis contaminates a fair amount of early topology because R and Rn are REALLY NICE, they are Finite Dimensional Metric Spaces which makes them pretty much literally every single thing simultaneously, and a lot of results that depend on this are used unnecessarily off the bat (which I view on the same lens as using the basis in Linear Algebra when it's not actually needed, nuclear cannons that harm intuition by just blowing the result and all of it's surrounding naturality).

    Yes, the notion of the definition of open and the regularity/normality and T spaces loses the motivation... partially, but on the other hand it forces you to build a much stronger intuition. A viable path is understanding open sets as the capacity to distinguish points and chunks of the space (which doesn't require the metric approach of talking about closeness and neighborhoods), from there it makes sense to start building the categories up to metric where one can distinguish not only points but it's possible to completely separate any pair of closed set.

    It's not a trivial path but IMO it's much more healthy to intuition to go to Analysis after Topology since you don't have the crutches of being able to just pull the metric and force the result (or even worse pull the basis and force the result). And Topology still covers Metric Spaces which are most of the treatment of Rn (most of the topology proof techniques for metric spaces are the Rn proofs but in topology language)

    Quote Originally Posted by PirateMonk View Post
    Also, analysis is not actually just the topology of R^n. It's true that you need a certain amount of topology to do analysis, so introductory courses spend a fair amount of time developing that, but most of the meat of analysis is in genuinely analytic concepts like differentiation, integration, series, and debatably uniform continuity, uniform convergence, and completeness, which do not generalize to arbitrary topological spaces.
    Not saying to skip analysis completely but to do Topology first to get all of the nice things done and just go down to Rn for the particulars.

    Fair amount is a bit short, you are practically only doing topology while occasionally proving results for metric spaces (hopefully not using Rn properties so the proof still holds when ignoring the finite dimensionality) until you reach the later chapters which deals with differentiation. Integration just needs a measure unless you mean Riemann Integration which I never see covered in Analysis but in Calculus for the introduction and then in Measure Theory for the killing blow after one shows that it's just a particular case of the Lebesgue integral (in the definite case). Uniform Continuity and Convergence are Metric affairs and Completeness is THE most metric of properties. A space can stop being complete after a change of metric (Rn is safe only because of the norm equivalence and this safety is also a trap that one has to unlearn eventually).

    It's not like Topology books outright ignore Metric Spaces just because Analysis covers a part of them.

    Quote Originally Posted by PirateMonk View Post
    Furthermore, the definition of a topological space is pretty hard to motivate unless you spend a certain amount of time playing with metric spaces (or do something like this). Yes, Alegh claims to like abstraction, but there's a difference between normal person liking abstraction and the kind which lets you jump into most topology textbooks with no background.
    One can never know until one tries it. It's harder to motivate but it's possible (there exists at least one such perspective) and particularizes to metric spaces later on.

    Quote Originally Posted by PirateMonk View Post
    As far as analysis books, I used Lebl's Basic Analysis for my first course. I'm not sure it's the best book, but it's better than many and free. If you want more abstraction, read chapter 7 early.
    I will second the Lebl (specially starting from Chapter 7 if approaching Analysis from Topology). A good free book as a stepping stone from Calculus to Analysis is Trench's on the other hand.
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    Default Re: Good book for self-study of undergraduate-level (math) analysis?

    Thanks for all of the recommendations! (Keep them coming if anyone has more - I probably won't actually have time to read through much in the way of things that require genuine concentration, let alone work exercises without getting interrupted all the time, until Winter Break.)

    The Topology versus Analysis argument is interesting. It reminds me of the time when I had a proof on a take-home exam in Finite Geometry, and I pointed out that the thing I was supposed to prove on the exam was actually true for all matrices via some things commonly proved in Linear Algebra (which I cited as my source in my proof - open-book take-home exam, remember) rather than anything I had to use special properties of the Affine Plane to prove just because I was supposed to be proving that it held in that particular case at the time. (I still feel I should have gotten full points for that answer.)

    I'll have to see which approach makes the most sense to me as an entry point as I try things, and how far I can get before it becomes clear that I need the structure of actual classes so I have someone to keep me on track (at which point I have to figure out which local school is the best place to take those classes). Ultimately, the only reason I need to take any undergrad-level classes is as prerequisites for the grad-level ones, and I don't know how hard math departments actually gatekeep against random people signing up to take grad-level math classes as a non-degree-seeking student. I would imagine they don't need to spend nearly as much time chasing off interested members of the public as, say, the art department does, so how many (and which) undergrad classes I need to take is probably more a question of which preparation gaps I have and can't fill in through self-study than anything else.

    (My favorite math "class" in undergrad was actually a non-credit weekly strategy/practice session the department ran for people planning on competing in the Putnam Exam to try a bunch of past exam problems and talk about strategies to try. I love figuring out that kind of stuff, particularly when you have to actually figure out an approach on your own because it's not directly connected to what someone was just telling you about in lecture, and I wish someone had shown me that kind of math back when I was an elementary school kid who hated the weekly timed multiplication tests because they wouldn't let me skip around and fill in the easy problems first.)

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    Default Re: Good book for self-study of undergraduate-level (math) analysis?

    If you want some good teaching with a dash of humour I can recommend the Cartoon Guides (especially The Cartoon Guide to Statistics).

    They are probably not as scholarly as the above, but are a fairly painless introduction to some of the basics. And, as a bonus, have lots of references for further reading.
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    Default Re: Good book for self-study of undergraduate-level (math) analysis?

    Take a look at Stein and Shakarchi's series on Fourier analysis, complex analysis, real analysis, and functional analysis. As an applied mathematics grad student with a weak background in computational maths in general, these books carried me through my undergraduate years.

    I would not bother with topology until starting real real analysis, i.e. measure theory and distributions, since qual exams in analysis are typically much harder than ones in topology, but that's up to your personal interests really. My introduction to topology was complex manifolds which really adds to my current research area, but your mileage may vary.

    I've done research in quantitative finance, Riemannian geometry, stochastic calculus, and information-theoretic cryptography. I also know a bit about quantum field theory and prelim-level microeconomics. I'm currently trying to learn algebraic geometry, but it's been hard.

    If you have any further questions about any of these, PM'd or otherwise, I'd be happy to answer.

    Also, @Rudin is brutal, it's not. The questions are brutally hard, and a good way to prepare for a top-tier school's quals. The writing and reasoning is astonishingly quick and clear, and it's a bit dense to read as a result, but the only times I've gotten stuck was when I was being stupid, rather than being tripped up on the author's mistakes. It's definitely a relief to be able to trust the author not to make major or minor errors, and I recommend it.

    But if Rudin's not an option, look at Pugh's. It's used at Berkeley and it's pretty good, better than Rudin some (not me) would say.

    My unpopular opinion though is students should skip undergrad real analysis and just do Fourier, complex, then measure theory.

    Edit: Also, regarding the last point you made, take as many grad classes as possible and try to join an REU or do research in another field like engineering (trust me, it's super easy to pick up with sufficient math knowledge). Ugrad classes were a waste of time, and despite my being an absolutely terrible student (because I was genuinely slow, in the mental sense), between grad classes and research I got into a top program.
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    Default Re: Good book for self-study of undergraduate-level (math) analysis?

    Which of Rudin's analysis texts are we talking about? Rudin's Real & Complex Analysis is pretty brutal. Principles of Mathematical Analysis, not so much.

    Also, yes, the Princeton Lectures in Analysis is a wonderful series. It makes a great companion to other texts, too.

    Another book that makes a nice second or third book on analysis or companion to such is General Theory of Functions and Integration by Angus E. Taylor.
    Last edited by gomipile; 2017-11-06 at 11:34 AM.
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    where is the atropal? and does it have a listed LA?

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    Default Re: Good book for self-study of undergraduate-level (math) analysis?

    Quote Originally Posted by rigsmal View Post
    I would not bother with topology until starting real real analysis, i.e. measure theory and distributions, since qual exams in analysis are typically much harder than ones in topology, but that's up to your personal interests really. My introduction to topology was complex manifolds which really adds to my current research area, but your mileage may vary.
    There is no reason to delay topology at all, specially since it won't even touch those topics. It's pretty self contained, and it can carry you all the way up to Homotopy and Homology at which point you now have nukes ready to deploy into geometry and analysis.

    Quote Originally Posted by rigsmal View Post
    Also, @Rudin is brutal, it's not. The questions are brutally hard, and a good way to prepare for a top-tier school's quals. The writing and reasoning is astonishingly quick and clear, and it's a bit dense to read as a result, but the only times I've gotten stuck was when I was being stupid, rather than being tripped up on the author's mistakes. It's definitely a relief to be able to trust the author not to make major or minor errors, and I recommend it.
    I refer to Papa Rudin. In the first chapters it has a habit of not skipping but outright vaulting over steps, if this is your first introduction to many of the concepts it covers there it's dense and unrewarding, it takes you so much time for so little progress (granted one can start with Baby Rudin and then go to Papa Rudin). Afterwards it's pretty much home free.

    Quote Originally Posted by rigsmal View Post
    My unpopular opinion though is students should skip undergrad real analysis and just do Fourier, complex, then measure theory.
    Fourier is so dependant in Real Analysis I find it better to outright ignore it until one does Functional Analysis to be able to reach it with a jackhammer obtained through far more pleasant means. Otherwise it becomes the longest most tedious exercise in massaging epsilons and deltas over and over and over slightly changing conditions that totally break the previous proofs.

    How do you even do Complex Analysis without Real Analysis (outside of just assuming all the results at which point why not just assume all of complex analysis as well). You end up needing Real Analysis to do it properly. Geometry can only carry you so far before you are waving your hands (unless you do Differential Surfaces or/and Riemannian Manifolds beforehand but those require Real Analysis for pretty much everything since the modus operandi is bring the surface back to R^n and clobber it with Real Analysis and then throw it's mangled body back up there with the Atlas). While Complex Analysis can start lifting on its own rather fast you need real analysis results from the get go (and you are periodically returning to it to poach the order of the reals or the fact that convexity and connectedness are one in the Reals).
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    Default Re: Good book for self-study of undergraduate-level (math) analysis?

    Quote Originally Posted by AsteriskAmp View Post
    Fourier is so dependant in Real Analysis I find it better to outright ignore it until one does Functional Analysis to be able to reach it with a jackhammer obtained through far more pleasant means. Otherwise it becomes the longest most tedious exercise in massaging epsilons and deltas over and over and over slightly changing conditions that totally break the previous proofs.

    How do you even do Complex Analysis without Real Analysis (outside of just assuming all the results at which point why not just assume all of complex analysis as well). You end up needing Real Analysis to do it properly. Geometry can only carry you so far before you are waving your hands (unless you do Differential Surfaces or/and Riemannian Manifolds beforehand but those require Real Analysis for pretty much everything since the modus operandi is bring the surface back to R^n and clobber it with Real Analysis and then throw it's mangled body back up there with the Atlas). While Complex Analysis can start lifting on its own rather fast you need real analysis results from the get go (and you are periodically returning to it to poach the order of the reals or the fact that convexity and connectedness are one in the Reals).
    1) When it comes to Fourier, I first did soft, non-rigorous Fourier, with some discussion of tempered distributions, then after hitting functional I did straight up harmonic analysis.

    2) Believe it or not, I took a complex analysis class using Ahlfors' book before I took any other math class save for calculus, linear algebra, Fourier, and an algebra class on group and ring theory. I was, and probably still am, very bad at math, but I managed it just fine then.

    When I first took ugrad analysis, it was pretty trivial because of the class curve, and I found the class to have taught me very little of consequence that Ahlfors and Fourier hadn't prepared me for. I wish I just jumped straight into real analysis with measure theory and distributions rather than waste time there. To its defense, I suppose it did build my skills at finding deltas, though I haven't really used that skill since.

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    Default Re: Good book for self-study of undergraduate-level (math) analysis?

    I just remembered a '90s video lecture of Walter B. Rudin's that is available in YouTube: "Set Theory: An Offspring of Analysis." I highly recommend it. He does a nice job of conceptually tying together a few of the things we've been discussing in this thread. It's not very technical, I think all of us could benefit from it.
    Quote Originally Posted by Harnel View Post
    where is the atropal? and does it have a listed LA?

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